 Solution

With $p_{-1}=1,p_0=a_0$, $q_{-1}=0,q_0=1$ and $p_n=a_n{p}_{n-1}+p_{n-2}$, $q_n=a_n{q}_{n-1}+q_{n-2}$, induction gives $p_n{q}_{n-1}-p_{n-1}{q}_n=(-1{)}^{n+1}$. Matrix multiplication gives $[a_0,\dots ,a_n,\beta ]=(p_n\beta +p_{n-1})/(q_n\beta +q_{n-1})$; monotonicity in $\beta >0$ places it strictly between the adjacent convergents.